Regge-Wheeler equation, linear stability, and greybody factors for dirty black holes

So-called "dirty'' black holes are those surrounded by nonzero stress energy, rather than vacuum. The presence of the nonzero stress energy modifies key features of the black hole, such as the surface gravity, Regge-Wheeler equation, linear stability, and greybody factors in a rather nontrivial way. Working within the inverse-Cowling approximation, (effectively the test-field limit), we shall present general forms for the Regge-Wheeler equation for linearized spin 0, spin 1, and axial spin 2 perturbations on an arbitrary static spherically symmetric background spacetime. Using very general features of the background spacetime, (in particular the classical energy conditions for the stress energy surrounding the black hole), we extract several interesting and robust bounds on the behavior of such systems, including rigorous bounds on the greybody factors for dirty black holes.